Mathematics Question and Solution
Calculating the indicated angle.
Let it be x.
Let 1 unit be the radius of the congruent circles each.
a = ⅕*180(5-2)
a = ⅕*180*3
a = 108°
a is the single interior angle of the inscribed regular pentagon.
1² = 2b²-2b²cos108
1 = 1.61803398875b
b = 1/1.61803398875
b = 0.61803398875 units.
b is the side length of the inscribed regular pentagon.
c = a-½(180-a)+60
c = 108-½(180-108)+60
c = 108-½(72)+60
c = 108-36+60
c = 72+60
c = 132°
d² = 1²+0.61803398875²-2*1*0.61803398875cos132
d = 1.48628965095 units.
Therefore x, the required indicated angle is;
x = 2e
Calculating e.
sine = ½(b)/d
sine = 0.5(0.61803398875)/1.48628965095
e = asin(0.5(0.61803398875)/1.48628965095)
e = 12°
It implies;
x = 2e
x = 2(12)
x = 24°
